Philosophy of Mathematics

What kind of discipline is mathematics?

We claim, perhaps controversially, that mathematics is not itself one of the natural

Note that a question like this about the classification of intellectual disciplines appears to be a philosophical question itself.

We support this claim with  three kinds of evidence:

History of mathematics vs. history of science

My remarks here will be more controversial than those under the headings that follow, and certainly do not agree with your course notes. I view natural science, properly so called, as beginning in Western Europe within the last 400 years. The earlier natural philosophy of the classical Greeks seems to me to differ from the modern natural sciences in essential attitude (modern science is based on self-conscious observation and (where possible) experiment in a way not seen earlier).

Mathematics is of course far older. There is no essential difference between mathematics as practiced by the ancient Greeks and mathematics as we practice it except the vast extension of the subject matter which has occurred recently. (This is itself a bold claim to which one could take exception: there are differences, such as the extension of rigorous methods to arithmetic as well as geometry and the modern attitude toward the actual infinite).

An amusing historical question: given the role of the Pythagorean school in the foundation of Greek mathematics, is mathematics a contemplative religion?

The Objects of Mathematics

The traditional objects of mathematics (divided further into arithmetic and geometry) were numbers and space. Initially, it was not clear that number was not to be identified with the objects of science: the Pythagoreans seem to be stating an atomic theory of physical reality when they say that "all is number". However, the crisis of the square root of two discredited Pythagorean atomism and seemed to make it clear that number at least was not to be identified with aspects of physical reality.

It took longer for it to be recognized that the geometry studied in mathematics is not to be identified with the geometry of physical space. Right into the 19th century, Euclidean geometry could be regarded (as by Kant) as a discipline which revealed logically necessary truths about the actual space of the world. The discovery of non-Euclidean geometry cast doubt on this view of the relationship between the space of geometry and the space of the physical world, and the observational successes of the theory of general relativity led most scientists to conclude that the geometry of Euclid was not the geometry of the physical world.

The further objects of mathematics at the time of the discovery of non-Euclidean geometry were numbers (of many varieties, some of which caused philosophical disquiet: rational and irrational, positive and negative, real and complex) and the functions which were the object of study in calculus. These had possible physical interpretations (though there were doubts about the physical meaning of, for example, imaginary numbers). At about the same time that non-Euclidean geometry was discovered, mathematicians began to investigate further abstractions, such as higher-dimensional geometries and hypercomplex numbers, whose connection to the physical world was clearly more remote.

The further enterprise of the late 19th and early 20th century in developing the foundations of mathematics in terms of set theory made it clear that the ontology of mathematics was not based on physical reality in any obvious way.

The Methods of Mathematics

The difference between the methods of mathematics and the methods of the natural sciences is glaring. The natural sciences rely on observation, and all results of natural science are in principle tentative, subject to refutation by further experiment. Mathematics proceeds by rigorous deduction from axioms whose truth is not subject to debate to conclusions whose truth is likewise not subject to debate -- unless there is an error in the reasoning that leads to the conclusions. The knowledge of the sciences, even if obtained correctly, is contingent and a posteriori whereas mathematical knowledge, if obtained correctly, appears to be a priori knowledge of necessary truths.

Other Views

There are certain vulnerabilities in my account. The historical argument can be challenged by considering the fact that of course there was natural philosophy prior to the flowering of unmistakably modern natural science in Western Europe, and there is some evidence of observation and experiment on the part of earlier workers. But it would be possible for me to grant that natural science is the same age as mathematics without granting that mathematics is one of the sciences.

Even my discussion of the objects of mathematics makes it clear that the objects (especially geometrical) of mathematics were clearly confused with aspects of physical reality until recently, and there are those who maintain that the abstract turn of mathematics in the late 19th century was a mistake and the real mathematics is that which models physical phenomena. Unfortunately for this argument, one of the most active areas of applied mathematics nowadays is exactly the pure mathematics of the foundations (logic and aspects of set theory) which turns out to be very useful in computer science.

The strongest point in my argument is the clear difference in method between mathematics and science. Historically, arithmetic and its offspring algebra and calculus were not pursued with the same logical rigor as geometry had been since the time of the Greeks. Mathematicians put this right in the late 19th and early 20th centuries: not only was arithmetic put on a firm logical foundation, but geometry was explained in terms of arithmetic (an outcome whose possibility was foreshadowed by the algebraic geometry of Descartes). This would have surprised the Greeks. The fact that arithmetic was then further explained in terms of set theory would have been even more surprising to them.

The significance of mathematics for philosophy of science

Whether natural science includes mathematics or not, it certainly depends on it. If the picture I have outlined of the relation between mathematics and science is correct, this should be a little disturbing for scientists, for a couple of reasons.

Mathematics is a rationalist discipline apparently studying a realm of "abstract" objects and engaged in determining necessary truths about these objects. Natural science is an empiricist discipline engaged in studying objects with which we are only acquainted by observation and about which we do not claim (in our more careful moments) to be determining necessary truths. This seems like a mismatch. In practice, it is enormously successful. Why? (This is more a question about philosophy of science)

The objects of mathematics have only an abstract nodding acquaintance with the objects of physics (at best we may identify some mathematical objects as idealizations of physical objects: think about elementary school level discussions of the difference between the circle we draw on paper and a real circle). The official ontology of mathematics explains all mathematical objects in terms of sets, which certainly do not seem to be physical objects. Geometric objects (and perhaps the functions of calculus) are actually fairly readily confused with (and often successfully understood in terms of) physical models: but this is not true of numbers. The number 17 seems to be something we learn about successfully in school, and so is the square root of negative one. But what are they? If they are not physical objects in the everyday world, what are they and how do we know about them? If there are no such objects, how do we succeed in talking about them (and, even more embarrassingly, in talking about them usefully?) This is more a question about philosophy of mathematics.

We do some philosophy of mathematics?

We will discuss some bits of mathematics with philosophical import. I might not cover all of these bits, depending on how much time we have.

The Pythagorean Scandal

I will briefly outline Pythagorean atomism (as far as we can now understand it). Everything was to be explained as number (and this means, as it always seemed to mean for the Greeks, positive whole number).

For Pythagorean atomism to work successfully, all lengths needed to be commensurable (I'll explain what this means). They discovered (and I will prove) that the length of the diagonal of a unit square is not commensurable with the lengths of its sides. They concluded on the basis of this logical proof that their view of reality was wrong (in this lies their greatness!) and thereafter the Greeks, who did not understand real number as we do, preferred geometry to arithmetic.

The Paradoxes of Zeno (and modern variations)

I'll talk about paradoxes of the infinite: two paradoxes of Zeno and Benardete's modern (but very similar and in some ways nicer) "paradox of the gods". I'll also mention some paradoxes of the infinite known in the Middle Ages and early modern times: the paradox of the lengths of concentric circles and the paradox of Galileo. These look forward to the work of Cantor on the sizes of infinite sets. The attitude of most philosophers and mathematicians until Cantor was that there is no actual infinity: all infinity is "potential". But St. Augustine is notable for saying something different quite early on.

The scandal of infinitesimals

I could talk briefly about a naive account of the basics of the calculus (how fast is a falling stone falling at a particular moment?). The naive account is clearly logically unsound (Bishop Berkeley was delighted to point this out). The scandal is perhaps not that the method is logically unsound (though mathematicians felt it necessary to fix this eventually, by replacing the use of infinitesimal numbers with the use of limits) but that it evidently worked (and later on (in the 1960's) the account in terms of infinitesimals was put on a correct logical footing). I gather that physicists and engineers still like the infinitesimals and are not too worried about the logical niceties.

What is a number?

In which sets make a first appearance: I will discuss the definition of counting numbers in terms of sets due to Frege and popularized by Bertrand Russell. Then I will confuse the issue entirely by revealing the modern definition of sets as numbers. Which one is right? Does it mean anything to ask which one is right? Does it matter?

I may further briefly outline the definition of real numbers due to Dedekind (restricting myself to positive real numbers to minimize technical complexities).

The point of this bit is to illustrate by example what it means to reduce mathematics (in this case arithmetic) to set theory. As soon as the real numbers are explained in terms of sets, notice that points in the usual Euclidean plane can be identified with pairs of real numbers using the older (but philosophically analogous) reduction of geometry to the arithmetic of the real numbers by Descartes (problematic when it was first proposed because geometry was at that time on a firmer logical footing than the real numbers!)

How many real numbers are there?

We discuss how many rational numbers there are and how many real numbers there are. I might briefly discuss the present status of Cantor's Continuum Hypothesis: the mathematical facts of the case are far beyond any reasonable possibility of discussion, but a brief account of what is known is of considerable philosophical interest.

Trouble with sets: Russell's paradox

I will briefly discuss Russell's paradox and the "crisis of foundations" (mostly to dismiss it; the solutions to the paradoxes are principled rather than ad hoc, in spite of what mathematicians tend to say about them, and there is no need for new ones (though there is technical and philosophical interest in the fact that there are a number of approaches that work).